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Quadratic Function

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Quadratic Function

Study Guide

Key Definition

A quadratic function is a polynomial function of degree 2, which can be expressed as $f(x) = ax^2 + bx + c$.

Important Notes

The graph of a quadratic function is a parabola.
The vertex of the parabola is the point where it changes direction.
When $a \gt 0$, the parabola opens upwards.
When $a \lt 0$, the parabola opens downwards.
The axis of symmetry is $x = -\frac{b}{2a}$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\sqrt{x}$ represents the square root of $x$

Remember to use proper notation when solving problems

Why It Works

Quadratic functions model parabolic paths in real-world situations, such as projectile motion, using the formula $f(x) = ax^2 + bx + c$.

Remember

The vertex form of a quadratic function is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

Quick Reference

Standard Form:$ax^2 + bx + c$

Vertex Form:$a(x-h)^2 + k$

Axis of Symmetry:$x = -\frac{b}{2a}$

Understanding Quadratic Function

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $f(x) = x^2$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the vertex of the quadratic function $y = (x-2)^2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder launches off a ramp. The height of his jump can be modeled by the equation $h(t) = -16t^2 + 32t + 5$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What are the roots of the quadratic equation $x^2 - 4x + 4 = 0$?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is the correct factorization of $x^2 + 5x + 6$?

Recap

Watch & Learn

Review key concepts and takeaways